# The class X student's school in krishnagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.(i) Taking $A$ as the origin, find the coordinates of the vertices of the triangle.(ii) What will be the coordinates of the vertices of $∆PQR$, if $C$ is the origin?Also, calculate the areas of the triangles in these cases. What do you observe?"

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Given:

The class X students school in krishnagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

To do:

We have to find :

(i) The coordinates of the vertices of the triangle.

(ii) The coordinates of the vertices of $∆PQR$, if $C$ is the origin and also, calculate the areas of the triangles in these cases. What do you observe.

Solution:

(i) Taking $A$ as the origin, the coordinates of the vertices of the $\triangle \mathrm{PQR}$ are $\mathbf{P}(4,6), \mathbf{Q}(3,2)$ and $\mathbf{R}(6,5)$.

(ii) Taking $\mathrm{C}$ as origin, the coordinates of the vertices of the $\triangle \mathrm{PQR}$ are $P(12,2), Q(13,6)$ and $R(10,3)$ Area of $\triangle \mathrm{PQR}$

$\mathrm{P}(4,6), \mathrm{Q}(3,2)$ and $R(6,5)$

Here, $x_{1}=4, y_{1}=6, x_{2}=3, y_{2}=2, x_{3}=6$ and $y_{3}=5$.

Area of $\triangle \mathrm{PQR}=\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]$

$=\frac{1}{2}[4(2-5)+3(5-6)+6(6-2)]$

$=\frac{1}{2}[-12-3+24]$

$=\frac{1}{2}[-15+24]$

$=\frac{1}{2}(9)$

$=\frac{9}{2}$ Hence, the area of $\triangle \mathrm{PQR}$ is $\frac{9}{2}$ sq units.

We observe that the area of $\triangle \mathrm{PQR}$ remains the same with different vertices.

Updated on 10-Oct-2022 13:22:09